Problem 138 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of succeeding terms. Find the series.

Step-by-Step Solution

Answer

The series could be either 3, -3, 3, -3, ... or 1, 1/3, 1/9, 1/27, ...

Step 1: Define the variables

Let the first term of the G.P. be \( a \) and the common ratio be \( r \). Then the second term is \( a \cdot r \), and so on.

Step 2: Form equations

From the given information, two equations can be made: \( a + a \cdot r = 5 \) (as the sum of first two terms is 5) and \( a = 3 \cdot (a \cdot r + a \cdot r^2) \) (as each term is three times the sum of its succeeding terms).

Step 3: Solve the equations simultaneously

Rearrange the second equation, we get \( a - 3a \cdot r = 3a \cdot r^2 \). Simplifying it, we get \( 1 -3r = 3r^2 \) or \( 3r^2 +3r -1 =0 \). This is a quadratic equation. Solve for \( r \). The solutions of the equation are \( r = -1 \) or \( r = 1/3 \). Put these values in the equation \( a (1 + r) = 5 \) to find the values of \( a \).

Step 4: Find the terms of the series

When \( r = -1 \), \( a = 3 \). The terms of the series are then 3, -3, 3, -3, .... When \( r = 1/3 \), \( a = 1 \). The terms are then 1, 1/3, 1/9, 1/27, ...

Key Concepts

Sum of Geometric Series

A geometric series is a sequence where each term is a constant multiple, called the common ratio, of the previous term. The sum of a geometric series, which can be finite or infinite, is an essential concept in mathematics.

An infinite geometric series refers to a sequence that continues indefinitely without end. For such series, the sum can converge (reach a specific number) only if the common ratio is between -1 and 1, excluding -1 and 1 themselves. If the absolute value of the common ratio is greater than 1, the series' sum would be infinite, as the terms would increase or decrease without bound.

In the provided problem, we are dealing with an infinite geometric series where the sum of the first two terms is given as 5. Using this information with the formula for the sum of the first two terms: \[ S_{2} = a + ar = 5 \]Until now, you might have thought of geometric series as sets of numbers. Remember, they are powerful tools for representing and solving complex real-world problems.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, generally shaped in the form of \[ ax^2 + bx + c = 0 \]where \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \).

In this exercise, the quadratic equation arises from manipulating the given relationships between terms in the geometric progression. We form the equation \[ 3r^2 + 3r - 1 = 0 \]

To find the common ratio \( r \), we solve this quadratic equation. Depending on the specific problem, we use different methods like factoring, completing the square, or using the quadratic formula, which is written as:\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Understanding quadratic equations is pivotal in solving geometric series problems, as it helps to derive critical values like the common ratio. This crucial step allows us to find the series' terms.

Common Ratio

The common ratio in a geometric progression is a fundamental concept that dictates the series' nature. It is defined as the constant factor between consecutive terms, expressed as:

\[ r = \frac{a_{n+1}}{a_{n}} \]

The common ratio can drastically change the series' behavior. If \( |r| < 1 \), the terms get progressively smaller, eventually converging to zero in the case of an infinite series. However, if the common ratio is greater than 1 or less than -1, it typically leads to terms growing without limit or alternately switching, as seen in oscillating series.

• If \( r = 1/3 \), our series terms shrink to smaller fractions.
• If \( r = -1 \), the series alternates between positive and negative terms.

Identifying the appropriate common ratio is vital for understanding how each term in the series relates to its neighbors and for efficiently solving associated problems.